We present zoom-in, AMR, high-resolution ($\simeq 30$ pc) simulations of high-redshift ($z\simeq 6$) galaxies with the aim of characterizing their internal properties and interstellar medium. Among other features, we adopt a star formation model based on a physically-sound molecular hydrogen prescription, and introduce a novel scheme for supernova feedback, stellar winds and dust-mediated radiation pressure. In the zoom-in simulation the target halo hosts "Dahlia", a galaxy with a stellar mass $M\ast =1.6\times 1010$M$\odot$, representative of a typical $z\sim 6$ Lyman Break Galaxy. Dahlia has a total H2 mass of $108.5$M$\odot$, that is mainly concentrated in a disk-like structure of effective radius $\simeq 0.6$ kpc and scale height $\simeq 200$ pc. Frequent mergers drive fresh gas towards the center of the disk, sustaining a star formation rate per unit area of $\simeq 15 $M$_\odot$ yr$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ kpc$\mathrm{<mrow\; data-mjx-texclass="ORD">-2</mrow>}$. The disk is composed by dense ($n\gtrsim 25$ cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-3</mrow>}$), metal-rich ($Z \simeq 0.5 $ Z$\odot$) gas, that is pressure-supported by radiation. We compute the $158\mu$m [CII] emission arising from Dahlia, and find that $\simeq 95\%$ of the total [CII] luminosity ($L[CII]\simeq 107.5$ L$\odot$) arises from the H2 disk. Although $30\%$ of the CII mass is transported out of the disk by outflows, such gas negligibly contributes to [CII] emission, due to its low density ($n\lesssim 10$ cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-3</mrow>}$) and metallicity ($Z\lesssim 10-1$Z$\odot$). Dahlia is under-luminous with respect to the local [CII]-SFR relation; however, its luminosity is consistent with upper limits derived for most $z\sim 6$ galaxies.